L(s) = 1 | + (−1.36 − 0.366i)2-s + (1.73 + i)4-s + (1.5 + 0.866i)5-s + (−2.59 + 0.5i)7-s + (−1.99 − 2i)8-s + (−1.73 − 1.73i)10-s + (−0.866 + 0.5i)11-s + (1.5 + 0.866i)13-s + (3.73 + 0.267i)14-s + (1.99 + 3.46i)16-s + 3.46i·17-s − 6.92·19-s + (1.73 + 3i)20-s + (1.36 − 0.366i)22-s + (4.33 + 2.5i)23-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 + 0.5i)4-s + (0.670 + 0.387i)5-s + (−0.981 + 0.188i)7-s + (−0.707 − 0.707i)8-s + (−0.547 − 0.547i)10-s + (−0.261 + 0.150i)11-s + (0.416 + 0.240i)13-s + (0.997 + 0.0716i)14-s + (0.499 + 0.866i)16-s + 0.840i·17-s − 1.58·19-s + (0.387 + 0.670i)20-s + (0.291 − 0.0780i)22-s + (0.902 + 0.521i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.342879 + 0.497900i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.342879 + 0.497900i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.36 + 0.366i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.59 - 0.5i)T \) |
good | 5 | \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.866 - 0.5i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.5 - 0.866i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 + (-4.33 - 2.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.33 - 7.5i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + (7.5 + 4.33i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.59 + 4.5i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + (4.33 - 7.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-12.9 - 7.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4iT - 71T^{2} \) |
| 73 | \( 1 - 10.3iT - 73T^{2} \) |
| 79 | \( 1 + (2.59 - 1.5i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.866 - 1.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 17.3iT - 89T^{2} \) |
| 97 | \( 1 + (4.5 - 2.59i)T + (48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.47818335721524114851036407255, −9.853642201725904358848699900872, −8.894555827191252607500974925282, −8.387282796139102033383649574767, −6.92389502732016569875441528263, −6.58902975377835463624364613258, −5.56635856714428223629430058577, −3.81689470192943826932689519201, −2.79557132676329502095916191083, −1.69219405540070865016033986734,
0.39675497382747421589457457038, 2.03026746601353201996401001446, 3.20445848258021719601532402300, 4.84216476282678967518926252717, 6.03663304531640849620653186782, 6.49291423754908361695778921656, 7.57705733235018519250845117708, 8.521052972684259456327897723810, 9.312389230824993167828009151724, 9.854527429200666762836276365839